New cost terms through homogenization of an optimal control problemunder dynamic boundary conditions on microscopic particles

This article concerns optimal control problems in a heterogeneous body with a periodical structure of particles depending on a small parameter \(\varepsilon\). We study the asymptotic behavior, as \(\varepsilon \to 0\), of the optimal control functional and of the optimal state when the initial problem is of parabolic type. We assume a dynamic condition and the effect of some controls for some of the particles on the boundary. In the so-called "critical case", we show the appearance of some new non-local in time "strange terms", not only in the limit parabolic equation but also in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear when controls are applied to the entire set of particles, or when the boundary condition on the particles is of Robin type. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/18/abstr.html